本次共计算 1 个题目：每一题对 x 求 4 阶导数。
    注意，变量是区分大小写的。
\[ \begin{equation}\begin{split}【1/1】求函数sin(arcsin(x)) 关于 x 的 4 阶导数：\\\end{split}\end{equation} \]\[ \begin{equation}\begin{split}\\解:&\\ &\color{blue}{函数的第 1 阶导数：}\\&\frac{d\left( sin(arcsin(x))\right)}{dx}\\=&cos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})\\=&\frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 2 阶导数：} \\&\frac{d\left( \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(arcsin(x)) + \frac{-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 3 阶导数：} \\&\frac{d\left( \frac{xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xcos(arcsin(x)) + \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} + \frac{x*-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{cos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{3x^{2}cos(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2xsin(arcsin(x))}{(-x^{2} + 1)^{2}} - \frac{cos(arcsin(x))}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\\\ &\color{blue}{函数的第 4 阶导数：} \\&\frac{d\left( \frac{3x^{2}cos(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{cos(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{2xsin(arcsin(x))}{(-x^{2} + 1)^{2}} - \frac{cos(arcsin(x))}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\right)}{dx}\\=&3(\frac{\frac{-5}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{7}{2}}})x^{2}cos(arcsin(x)) + \frac{3*2xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} + \frac{3x^{2}*-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{5}{2}}} + (\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})cos(arcsin(x)) + \frac{-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}} - \frac{(\frac{\frac{-3}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{5}{2}}})xsin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{xcos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - 2(\frac{-2(-2x + 0)}{(-x^{2} + 1)^{3}})xsin(arcsin(x)) - \frac{2sin(arcsin(x))}{(-x^{2} + 1)^{2}} - \frac{2xcos(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)^{2}} - \frac{(\frac{-(-2x + 0)}{(-x^{2} + 1)^{2}})cos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}} - \frac{(\frac{\frac{-1}{2}(-2x + 0)}{(-x^{2} + 1)^{\frac{3}{2}}})cos(arcsin(x))}{(-x^{2} + 1)} - \frac{-sin(arcsin(x))(\frac{(1)}{((1 - (x)^{2})^{\frac{1}{2}})})}{(-x^{2} + 1)(-x^{2} + 1)^{\frac{1}{2}}}\\=&\frac{15x^{3}cos(arcsin(x))}{(-x^{2} + 1)^{\frac{7}{2}}} + \frac{6xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{5}{2}}} - \frac{3x^{2}sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{5}{2}}} - \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{\frac{3}{2}}} - \frac{12x^{2}sin(arcsin(x))}{(-x^{2} + 1)^{3}} - \frac{3sin(arcsin(x))}{(-x^{2} + 1)^{2}} - \frac{xcos(arcsin(x))}{(-x^{2} + 1)^{2}(-x^{2} + 1)^{\frac{1}{2}}} - \frac{2xcos(arcsin(x))}{(-x^{2} + 1)^{\frac{1}{2}}(-x^{2} + 1)^{2}} + \frac{sin(arcsin(x))}{(-x^{2} + 1)^{\frac{3}{2}}(-x^{2} + 1)^{\frac{1}{2}}}\\ \end{split}\end{equation} \]

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